Plasmon excitations of single-wall carbon nanotubes
S. Dmitrović,* T. Vuković, B. Nikolić, M. Damnjanović, and I. Milošević
Faculty of Physics, University of Belgrade, P.O. Box 368, Belgrade 11001, Serbia
Received 22 January 2008; revised manuscript received 2 May 2008; published 11 June 2008
Plasmon excitations in isolated single-wall carbon nanotubes are singled out in the optical spectra, and
analyzed within density-functional tight-binding method and random-phase approximation. Full symmetry
considerations, implemented in both approaches, stressed out that only helical quantum numbers are conserved
in processes involving momentum transfer, i.e., only these quantum numbers can be unambiguously attributed
to the plasmons. Energy of plasmon is about 5 eV and slightly decreases with diameter with no observable
influence of chirality. Energy of + plasmon is between 20 and 21 eV, and is insensitive to nanotube
geometry. Dispersion of the plasmon is mainly linear. Slope of dispersion curve increases with tube’s
translational period.
DOI: 10.1103/PhysRevB.77.245415 PACS numbers: 61.46.Fg, 78.67.Ch, 71.45.Gm, 73.20.Mf
I. INTRODUCTION
Being one of the most important nanotechnological mate-
rials, carbon nanotubes are examined in many details.
1
Still,
there is a lack of complete insight to the collective electronic
excitations plasmons, as the predictions in the existing
studies differ due to the theoretical models applied.
Experimentally, electron energy-loss spectroscopy
EELS is the main tool for plasmon investigation in crystals.
For single-wall carbon nanotubes SWCNTs, measurements
in both aloof and penetrating modes have been performed.
2–5
Reported plasmons of isolated SWCNT Refs. 4 and 5 are in
the low-loss region up to 50 eV: a single surface plasmon
at 15 eV and two volume-plasmon peaks around 5 and 23 eV.
The high energy plasmon is associated with collective exci-
tations of all valence electrons + plasmon while the
lowest one comes solely from valence electrons plas-
mon. One of the earliest measurements on SWCNT
bundles,
2
with tubes of 1.0–1.3 nm in diameter, stated
plasmon at energy of 5.8 eV and also the + plasmon at
20.6 eV. A single study
3
of momentum-dependent EELS for
parallel polarization exists in the literature. These measure-
ments are performed for bulk samples of purified SWCNTs,
with mean diameter of 1.4 nm, showing the nondispersive
excitations with energies below 4 eV and two dispersive ex-
citations, one at 5.2 eV and another at 21.5 eV, for momen-
tum transfer along the tubes’ axis. Prominent peak at 4.5 eV,
independent of the tube’s diameter, has also been obtained in
polarized optical-absorption measurements
6
on vertically
aligned SWCNTs for parallel polarization stimulating further
discussion about assignation of this peak to plasmon.
When it comes to the reported theoretical studies, only a
few of them goes beyond quasiclassical hydrodynamical
model. However, quantum-mechanical treatment is necessary
to encounter the effects of tube’s chirality, small wall thick-
ness, etc. Zone folding of the graphene tight-binding TB
electron bands had been used to calculate
7
dielectric matrix
within the random-phase approximation RPA mainly for
tubes with large diameter. Using a similar approach, but in-
cluding local-field effects,
8
Perez and Que in Ref. 8 con-
cluded that for chiral SWCNTs, the collective electronic
modes are dispersive while for the achiral tubes modes are
nondispersive. All of these studies are related solely to
plasmon. Only a single study
9
of both experimentally re-
ported volume plasmons is based on DFT calculations of
both optical and energy-loss spectra of isolated and aligned
SWCNTs with small diameters. These results are in excellent
agreement with measurements, showing also that parallel to
the tube axis, exchange-correlation and local-field effects
were dumped.
The aim of this paper is to thoroughly describe volume
plasmons behavior of individual SWCNTs of an arbitrary
geometry. Starting with full symmetry based and high-
precision density-functional tight-binding DFTB electronic
Bloch states, we use two different approaches,
10,11
optical
and RPA, as explained in Sec. II. The main results, i.e.,
and + plasmon energies and their dispersion properties,
and also comparison with the interband transitions are pre-
sented in Sec. III.
II. METHOD
Two different models accounted here are optical calcula-
tions of the transversal dielectric function with zero trans-
ferred momentum and RPA calculations of the longitudinal
dielectric function needed for EELS calculations. Electronic
energies and states are obtained by the full symmetry imple-
mented DFTB method code POLSYM
12
. Full symmetry
groups of SWCNTs Ref. 13 are line groups L
C
= Lq
p
22a
for chiral tubes and L
ZA
= L2n
n
/ mcma for achiral ones;
here, helicity p, and the orders q and n = GCDn
1
, n
2
= GCDq , p of the rotational axes of the isogonal group and
of the line group are uniquely determined by the chiral indi-
ces n
1
and n
2
, while the translational period a is found by
relaxation. This symmetry implies that the Bloch eigenfunc-
tions |km and the corresponding eigenenergies
k,m
are
assigned by the following quantum numbers: quasimomen-
tum k - / a , / a, z component of the angular momen-
tum m =-q / 2,-q / 2+1,..., q / 2, and parities
U
,
v
and
h
invoked by U axis and, for achiral tubes only, horizontal and
vertical mirror planes taking the values +1 for even and -1
for odd states while conventionally 0, otherwise. These
quantum numbers are employed to get severe selection rules,
significantly reducing the number of relevant matrix ele-
PHYSICAL REVIEW B 77, 245415 2008
1098-0121/2008/7724/2454156 ©2008 The American Physical Society 245415-1